pam

Partitioning Around Medoids

Partitioning (clustering) of the data into k clusters “around medoids”, a more robust version of K-means.

Usage

pam(x, k, diss = inherits(x, "dist"),
    metric = c("euclidean", "manhattan"), 
    medoids = NULL, stand = FALSE, cluster.only = FALSE,
    do.swap = TRUE,
    keep.diss = !diss && !cluster.only && n < 100,
    keep.data = !diss && !cluster.only,
    pamonce = FALSE, trace.lev = 0)

Details

The basic pam algorithm is fully described in chapter 2 of Kaufman and Rousseeuw(1990). Compared to the k-means approach in kmeans, the function pam has the following features: (a) it also accepts a dissimilarity matrix; (b) it is more robust because it minimizes a sum of dissimilarities instead of a sum of squared euclidean distances; (c) it provides a novel graphical display, the silhouette plot (see plot.partition) (d) it allows to select the number of clusters using mean(silhouette(pr)[, "sil_width"]) on the result pr <- pam(..), or directly its component pr$silinfo$avg.width, see also pam.object.

When cluster.only is true, the result is simply a (possibly named) integer vector specifying the clustering, i.e., pam(x,k, cluster.only=TRUE) is the same as pam(x,k)$clustering but computed more efficiently.

The pam-algorithm is based on the search for k representative objects or medoids among the observations of the dataset. These observations should represent the structure of the data. After finding a set of k medoids, k clusters are constructed by assigning each observation to the nearest medoid. The goal is to find k representative objects which minimize the sum of the dissimilarities of the observations to their closest representative object.

By default, when medoids are not specified, the algorithm first looks for a good initial set of medoids (this is called the build phase). Then it finds a local minimum for the objective function, that is, a solution such that there is no single switch of an observation with a medoid that will decrease the objective (this is called the swap phase).

When the medoids are specified, their order does not matter; in general, the algorithms have been designed to not depend on the order of the observations.

The pamonce option, new in cluster 1.14.2 (Jan. 2012), has been proposed by Matthias Studer, University of Geneva, based on the findings by Reynolds et al. (2006) and was extended by Erich Schubert, TU Dortmund, with the FastPAM optimizations.

The default FALSE (or integer 0) corresponds to the original “swap” algorithm, whereas pamonce = 1 (or TRUE), corresponds to the first proposal .... and pamonce = 2 additionally implements the second proposal as well.

The key ideas of FastPAM (Schubert and Rousseeuw, 2019) are implemented except for the linear approximate build as follows:

pamonce = 3:

reduces the runtime by a factor of O(k) by exploiting that points cannot be closest to all current medoids at the same time.

pamonce = 4:

additionally allows executing multiple swaps per iteration, usually reducing the number of iterations.

pamonce = 5:

adds minor optimizations copied from the pamonce = 2 approach, and is expected to be the fastest variant.

Note

For large datasets, pam may need too much memory or too much computation time since both are \(O(n^2)\). Then, clara() is preferable, see its documentation.

There is hard limit currently, \(n \le 65536\), at \(2^{16}\) because for larger \(n\), \(n(n-1)/2\) is larger than the maximal integer (.Machine$integer.max = \(2^{31} - 1\)).

References

Reynolds, A., Richards, G., de la Iglesia, B. and Rayward-Smith, V. (1992) Clustering rules: A comparison of partitioning and hierarchical clustering algorithms; Journal of Mathematical Modelling and Algorithms 5, 475--504. 10.1007/s10852-005-9022-1.

Erich Schubert and Peter J. Rousseeuw (2019) Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms; Preprint, (https://arxiv.org/abs/1810.05691).

See Also

agnes for background and references; pam.object, clara, daisy, partition.object, plot.partition, dist.

Aliases

• pam

Examples

# NOT RUN {
## generate 25 objects, divided into 2 clusters.
x <- rbind(cbind(rnorm(10,0,0.5), rnorm(10,0,0.5)),
           cbind(rnorm(15,5,0.5), rnorm(15,5,0.5)))
pamx <- pam(x, 2)
pamx # Medoids: '7' and '25' ...
summary(pamx)
plot(pamx)
## use obs. 1 & 16 as starting medoids -- same result (typically)
(p2m <- pam(x, 2, medoids = c(1,16)))
## no _build_ *and* no _swap_ phase: just cluster all obs. around (1, 16):
p2.s <- pam(x, 2, medoids = c(1,16), do.swap = FALSE)
p2.s

p3m <- pam(x, 3, trace = 2)
## rather stupid initial medoids:
(p3m. <- pam(x, 3, medoids = 3:1, trace = 1))

# }
# NOT RUN {
pam(daisy(x, metric = "manhattan"), 2, diss = TRUE)

data(ruspini)
## Plot similar to Figure 4 in Stryuf et al (1996)
# }
# NOT RUN {
plot(pam(ruspini, 4), ask = TRUE)
# }